Optimal. Leaf size=72 \[ \frac {2 e^{n \tanh ^{-1}(a x)}}{a c^2 n \left (4-n^2\right )}-\frac {e^{n \tanh ^{-1}(a x)} (n-2 a x)}{a c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )} \]
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Rubi [A]
time = 0.09, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6271, 6272}
\begin {gather*} \frac {2 e^{n \tanh ^{-1}(a x)}}{a c^2 n \left (4-n^2\right )}-\frac {(n-2 a x) e^{n \tanh ^{-1}(a x)}}{a c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 6271
Rule 6272
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx &=-\frac {e^{n \tanh ^{-1}(a x)} (n-2 a x)}{a c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )}+\frac {2 \int \frac {e^{n \tanh ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{c \left (4-n^2\right )}\\ &=\frac {2 e^{n \tanh ^{-1}(a x)}}{a c^2 n \left (4-n^2\right )}-\frac {e^{n \tanh ^{-1}(a x)} (n-2 a x)}{a c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 68, normalized size = 0.94 \begin {gather*} -\frac {(1-a x)^{-1-\frac {n}{2}} (1+a x)^{-1+\frac {n}{2}} \left (2-n^2+2 a n x-2 a^2 x^2\right )}{a c^2 (-2+n) n (2+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 55, normalized size = 0.76
method | result | size |
gosper | \(-\frac {{\mathrm e}^{n \arctanh \left (a x \right )} \left (2 a^{2} x^{2}-2 n a x +n^{2}-2\right )}{\left (a^{2} x^{2}-1\right ) c^{2} a n \left (n^{2}-4\right )}\) | \(55\) |
risch | \(-\frac {\left (2 a^{2} x^{2}-2 n a x +n^{2}-2\right ) \left (-a x +1\right )^{-\frac {n}{2}} \left (a x +1\right )^{\frac {n}{2}}}{\left (a^{2} x^{2}-1\right ) c^{2} a n \left (n^{2}-4\right )}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 80, normalized size = 1.11 \begin {gather*} \frac {{\left (2 \, a^{2} x^{2} - 2 \, a n x + n^{2} - 2\right )} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a c^{2} n^{3} - 4 \, a c^{2} n - {\left (a^{3} c^{2} n^{3} - 4 \, a^{3} c^{2} n\right )} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} \frac {x}{c^{2}} & \text {for}\: a = 0 \\\tilde {\infty } x e^{- \infty n} & \text {for}\: a = - \frac {1}{x} \\\tilde {\infty } x e^{\infty n} & \text {for}\: a = \frac {1}{x} \\- \frac {a^{2} x^{2} \operatorname {atanh}{\left (a x \right )}}{4 a^{3} c^{2} x^{2} e^{2 \operatorname {atanh}{\left (a x \right )}} - 4 a c^{2} e^{2 \operatorname {atanh}{\left (a x \right )}}} - \frac {2 a x \operatorname {atanh}{\left (a x \right )}}{4 a^{3} c^{2} x^{2} e^{2 \operatorname {atanh}{\left (a x \right )}} - 4 a c^{2} e^{2 \operatorname {atanh}{\left (a x \right )}}} + \frac {a x}{4 a^{3} c^{2} x^{2} e^{2 \operatorname {atanh}{\left (a x \right )}} - 4 a c^{2} e^{2 \operatorname {atanh}{\left (a x \right )}}} - \frac {\operatorname {atanh}{\left (a x \right )}}{4 a^{3} c^{2} x^{2} e^{2 \operatorname {atanh}{\left (a x \right )}} - 4 a c^{2} e^{2 \operatorname {atanh}{\left (a x \right )}}} + \frac {2}{4 a^{3} c^{2} x^{2} e^{2 \operatorname {atanh}{\left (a x \right )}} - 4 a c^{2} e^{2 \operatorname {atanh}{\left (a x \right )}}} & \text {for}\: n = -2 \\- \frac {a^{2} x^{2} \log {\left (x - \frac {1}{a} \right )}}{4 a^{3} c^{2} x^{2} - 4 a c^{2}} + \frac {a^{2} x^{2} \log {\left (x + \frac {1}{a} \right )}}{4 a^{3} c^{2} x^{2} - 4 a c^{2}} - \frac {2 a x}{4 a^{3} c^{2} x^{2} - 4 a c^{2}} + \frac {\log {\left (x - \frac {1}{a} \right )}}{4 a^{3} c^{2} x^{2} - 4 a c^{2}} - \frac {\log {\left (x + \frac {1}{a} \right )}}{4 a^{3} c^{2} x^{2} - 4 a c^{2}} & \text {for}\: n = 0 \\\frac {\int \frac {e^{2 \operatorname {atanh}{\left (a x \right )}}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} & \text {for}\: n = 2 \\- \frac {2 a^{2} x^{2} e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{3} c^{2} n^{3} x^{2} - 4 a^{3} c^{2} n x^{2} - a c^{2} n^{3} + 4 a c^{2} n} + \frac {2 a n x e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{3} c^{2} n^{3} x^{2} - 4 a^{3} c^{2} n x^{2} - a c^{2} n^{3} + 4 a c^{2} n} - \frac {n^{2} e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{3} c^{2} n^{3} x^{2} - 4 a^{3} c^{2} n x^{2} - a c^{2} n^{3} + 4 a c^{2} n} + \frac {2 e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{3} c^{2} n^{3} x^{2} - 4 a^{3} c^{2} n x^{2} - a c^{2} n^{3} + 4 a c^{2} n} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.21, size = 93, normalized size = 1.29 \begin {gather*} \frac {{\left (a\,x+1\right )}^{n/2}\,\left (\frac {2\,x^2}{a\,c^2\,n\,\left (n^2-4\right )}-\frac {2\,x}{a^2\,c^2\,\left (n^2-4\right )}+\frac {n^2-2}{a^3\,c^2\,n\,\left (n^2-4\right )}\right )}{\left (\frac {1}{a^2}-x^2\right )\,{\left (1-a\,x\right )}^{n/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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